L4

We frame dynamic persuasion in a partial observation stochastic control game with an ergodic criterion. The receiver controls the dynamics of a multidimensional unobserved state process. Information is provided to the receiver through a device designed by the sender that generates the observation process. 

The commitment of the sender is enforced and an exogenous information process outside the control of the sender is allowed. We develop this approach in the case where all dynamics are linear and the preferences of the receiver are linear-quadratic.

We prove a verification theorem for the existence and uniqueness of the solution of the HJB equation satisfied by the receiver’s value function. An extension to the case of persuasion of a mean field of interacting receivers is also provided. We illustrate this approach in two applications: the provision of information to electricity consumers with a smart meter designed by an electricity producer; the information provided by carbon footprint accounting rules to companies engaged in a best-in-class emissions reduction effort. In the first application, we link the benefits of information provision to the mispricing of electricity production. In the latter, we show that when firms declare a high level of best-in-class target, the information provided by stringent accounting rules offsets the Nash equilibrium effect that leads firms to increase pollution to make their target easier to achieve.

This is a joint work with Prof. René Aïd, Prof. Giorgia Callegaro and Prof. Luciano Campi.

In this talk we study the numerical approximation of the jump-diffusion McKean--Vlasov SDEs with super-linearly growing drift, diffusion and jump-coefficient. In the first step, we derive the corresponding interacting particle system and define a Milstein-type approximation for this. Making use of the propagation of chaos result and investigating the error of the Milstein-type scheme we provide convergence results for the scheme. In a second step, we discuss potential simplifications of the numerical approximation scheme for the direct approximation of the jump-diffusion McKean--Vlasov SDE. Lastly, we present the results of our numerical simulations.

On arbitrary Carnot groups, the only hypoelliptic Hodge-Laplacians on forms that have been introduced are 0-order pseudodifferential operators constructed using the Rumin complex.  However, to address questions where one needs sharp estimates, this 0-order operator is not suitable. Indeed, this is a rather difficult problem to tackle in full generality, the main issue being that the Rumin exterior differential is not homogeneous on arbitrary Carnot groups. In this talk, I will focus on the specific example of the free Carnot group of step 3 with 2 generators, where it is possible to introduce different hypoelliptic Hodge-Laplacians on forms. Such Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups

I will introduce the class of causally-null-compactifiable spacetimes that can be canonically converted into compact timed-metric spaces using the cosmological time function of Andersson-Galloway-Howard and the null distance of Sormani-Vega. This class of space-times includes future developments of compact initial data sets and regions exhausting asymptotically flat space-times. I will discuss various intrinsic notions of distance between such space-times and show that some of them are definite in the sense that they are equal to zero if and only if there is a time-oriented Lorentzian isometry between the space-times. These definite distances allow us to define notions of convergence of space-times to limit space-times that are not necessarily smooth. This is joint work with Christina Sormani.

We prove the local Lipschitz continuity of energy minimizing harmonic maps between singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.  Joint work with Yaoting Gui and Jürgen Jost.

A well-known problem in algebraic geometry is to construct smooth projective Calabi--Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi--Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov--Tian--Todorov theorem as well as its variant for pairs of a log Calabi--Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.
 

I want to give a survey about the rational cohomology of SL_n
Z. This includes recent developments of finding Hopf algebras in the
direct sum of all cohomology groups of SL_n Z for all n. I will give a
quick overview about Hopf algebras and what this structure implies for
the cohomology of SL_n Z.

I will present a notion of spin structure on a perfect complex in characteristic zero, generalizing the classical notion for an (algebraic) vector bundle. For a complex $E$ on $X$ with an oriented quadratic structure one obtains an associated ${\mathbb Z}/2{\mathbb Z}$-gerbe over X which obstructs the existence of a spin structure on $E$. This situation arises naturally on moduli spaces of coherent sheaves on Calabi-Yau fourfolds. Using spin structures as orientation data, we construct a categorical refinement of a K-theory class constructed by Oh-Thomas on such moduli spaces.

Bryant’s Laplacian flow is an analogue of Ricci flow that seeks to flow an arbitrary initial closed $G_2$-structure on a 7-manifold toward a torsion-free one, to obtain a Ricci-flat metric with holonomy $G_2$. This talk will give an overview of joint work with Mark Haskins and Rowan Juneman about complete self-similar solutions on the anti-self-dual bundles of ${\mathbb CP}^2$ and $S^4$, with cohomogeneity one actions by SU(3) and Sp(2) respectively. We exhibit examples of all three classes of soliton (steady, expander and shrinker) that are asymptotically conical. In the steady case these form a 1-parameter family, with a complete soliton with exponential volume growth at the boundary of the family. All complete Sp(2)-invariant expanders are asymptotically conical, but in the SU(3)-invariant case there appears to be a boundary of complete expanders with doubly exponential volume growth.